Absolute continuity and $α$-numbers on the real line

Abstract

Let $μ,ν$ be Radon measures on $\mathbb{R}$, with $μ$ non-atomic and $ν$ doubling, and write $μ= μ_{a} + μ_{s}$ for the Lebesgue decomposition of $μ$ relative to $ν$. For an interval $I \subset \mathbb{R}$, define $α_{μ,ν}(I) := \mathbb{W}_{1}(μ_{I},ν_{I})$, the Wasserstein distance of normalised blow-ups of $μ$ and $ν$ restricted to $I$. Let $\mathcal{S}_ν$ be the square function $$\mathcal{S}^{2}_ν(μ) = \sum_{I \in \mathcal{D}} α_{μ,ν}^{2}(I)χ_{I},$$ where $\mathcal{D}$ is the family of dyadic intervals of side-length at most one. I prove that $\mathcal{S}_ν(μ)$ is finite $μ_{a}$ almost everywhere, and infinite $μ_{s}$ almost everywhere. I also prove a version of the result for a non-dyadic variant of the square function $\mathcal{S}_ν(μ)$. The results answer the simplest "$n = d = 1"$ case of a problem of J. Azzam, G. David and T. Toro.